Differential equations have their origin in the quest to describe nature by mathematics, and these equations are perfect tools to describe physical phenomena that vary in space and time. In fact, all the fundamental laws of nature are given by differential equations: Newton’s law describes gravitation, Maxwell’s equations describe electromagnetism, Navier–Stokes’ equations describe fluid flow, etc. Thus, they model a wide variety of phenomena, and have been extensively studied by mathematicians, physicists, engineers, and others. Differential equations constitute a core mathematical discipline. However, several key questions have not been answered, or only been partly answered. Central issues are:

• Do the equations have a solution? It is still unknown if the equations that describe the flow of air around an airplane have a solution.
• Given that the equation has a solution, is it unique? For flow of oil in petroleum reservoirs, the equations give several possible distinct values for the saturation at the same point in space.
• Is the solution stable? Will a small change in one place in space have dramatic consequences in a completely different location (“the butterfly effect”).
• If the equation has a unique and stable solution, how can one determine the solution? Given the measured weather data for today, how can one predict the weather for tomorrow?

Variations of these questions for carefully selected nonlinear partial differential equations constituted the focus for the research program, in particular the interplay between theoretical results (mathematical properties of the solution) and numerical computations (how to compute the solution). Differential equations have been an intensive research area in mathematics in Norway the last two decades, and there are now strong research groups in Oslo, Trondheim, and Bergen. The research program further reinforced the strong national activities.

The program at CAS has given the participants a unique opportunity to work focused and uninterrupted on difficult problems, and to strengthen collaboration with foreign colleagues and commence new partnership for joint work in the future. 

Progress has been obtained along the following lines:

First of all the program has provided a stimulating atmosphere to complete or considerably advance projects already initiated before the program at CAS. Examples of this include work by Jenssen and Karper on one-dimensional flow with temperature dependent transport coefficients, and the completion of the monograph by Feireisl. Another example is the work by Coclite, Karlsen, and Kwon on the Degasperis-Procesi equation. Let me also mention the work by Andreianov, Bendahmane, Karlsen, and Ouaro on well-posedness results for nonlinear degenerate parabolic equations and convergence of finite volume schemes.

Secondly, the program has given a singular opportunity to commence new projects with new collaborators. Some of the problems surfaced during the program, while some had been discussed prior to the program. Here we can mention the work of Bressan with Holden and Raynaud regarding the Lipschitz metric for the Hunter–Saxton equation. We should also mention the work by Andreianov, Karlsen, and Risebro on the development of a new general theory for scalar conservation laws with discontinuous flux, and the work by Carrillo, Karper, and Trivisa on the dynamics of a fluid-particle interaction model in the bubbling regime, and Chen, Ding, and Karlsen’s results on multidimensional stochastic scalar conservation laws and degenerate parabolic equations.

Finally, Raynaud received a prize for young researchers from the Royal Norwegian Society for Sciences and Letters while at CAS.

The program has served as a seed for future research and collaborations that without doubt will prove to be highly fruitful in the years to come